基於李亞諾夫之穩定定理(Lyapunov Theorem) ，本論文針對具有非完整約束之擾動非線性系統，提出適應性可變結構追蹤控制器來解決軌跡追蹤問題。首先在第一階段，將非齊次動態約束之系統轉換成多鏈接形式，以及利用步階回歸控制技巧來設計一個想要的速度輸入函數。接著第二階段，藉由可變結構控制方法(VSC)設計具有調適機制的可變結構控制器，使得轉換的狀態追蹤誤差和原有的狀態追蹤誤差都能夠達到漸進穩定之性能指標。此外，控制系統之速度軌跡能夠在有限時間內追蹤得到在第一階段所設計之想要的速度輸入函數，進而達到狀態軌跡追蹤之目的。由於控制器採取適應機制，所以擾動的上界資訊不需要事先預知。最後，本論文提供一個數值範例及一個實際範例以驗證所提出控制器設計法則的可行性。

Based on the Lyapunov stability theorem, a design methodology of adaptive variable structure tracking controller is proposed in this thesis for a class of perturbed nonlinear systems with nonholonomic constraints to solve the trajectory tracking control problems. In the first stage, the nonhomogeneous dynamics constraints of the plant are transformed into a multi-chained form, and a desired velocity input function is designed by utilizing backstepping control technique. In the second stage, the proposed controller is designed by using variable structure control (VSC) methodology with adaptive mechanism embedded, so that tracking errors of the transformed states and original states are able to achieve asymptotic stability. In addition, the velocity trajectories of the controlled systems can track the desired velocity input function designed in the first stage in a finite time. The upper bounds of perturbations are not required to be known in advance due to employed adaptive mechanisms. Finally, a numerical and a practical example are given for demonstrating the feasibility of the proposed control scheme.

論文審定書 i
致謝 ii
中文摘要 iii
Abstract iv
List of Figures vii
List of Notations ix

Chapter 1 Introduction 1
1.1 Motivation 1
1.2 Brief Sketch of the Contents 3

Chapter 2 Nonholonomic Systems 5
2.1 System Descriptions and Affine Constraints 5
2.2 Expansion of the Vector Space Δ̅m 9
2.3 Transformation of the Nonholonomic Constraints into System in Multiple Chained form 13

Chapter 3 Design of the Control Scheme 19
3.1 Formulation of Dynamic Equations of Tracking Errors 19
3.2 Design of the Desired Velocity Input Function v* 23
3.3 Design of the Variable Structure Controller 33
3.4 Summary of Design Procedure 41

Chapter 4 Computer Simulation and Practical Application 42
4.1 Numerical Example 42
4.2 Practical Application 46

Chapter 5 Conclusions 62

Bibliography 63
Appendix A 70
Appendix B 71

[1] M. Kochemn, and R. Isermann, “Nonlinear path following control of
nonholonomic vehicles with the human as actuator in a parking application,”
International Journal of Robust and Nonlinear Control , vol. 14, no. 6, pp. 513-
524, 2004.
[2] A. M. Bloch, M. Reyhanoglu, and N. H. McClamroch, “Control and stabilization
of Nonholonomic Dynamic Systems,” IEEE Transactions on Automatic Control,
vol.37, no. 11, pp. 1746-1757, 1992.
[3] I. Kolmanovsky, and N. H. McClamroch, “Developments in nonholonomic
control problems,” IEEE Control System, vol. 15, no. 6, pp. 20-35, 1995.
[4] H. Krishnan, M. Reyhanoglu, and H. McClamroch, “Attitude stabilization of a
rigid spacecraft using two control torques: a nonlinear control approach based
on the spacecraft attitude dynamics,” Automatica, vol. 30, no. 6, pp. 1023-
1027, 1994.
[5] J. Ye, “Tracking control for nonholonomic mobile robots: integrating the analog
neural network into the backstepping technique,” Neurocomputing, vol. 71, no.
16-18, pp. 3373-3378, 2008.
[6] R. Postoyan, M. C. Bragagnolo, E. Galbrun, J. Daafouz, D. Nesic, and E. B.
Castelan,“Event-triggered tracking control of unicycle mobile robots,”
Automatica, vol. 52, pp. 302-308 , 2015.
[7] R. Zheng, Y. Liu, and D. Sun, “Enclosing a target by nonholonomic mobile
robots with bearing-only measurements,” Automatica, vol. 53, pp. 400-407,
2015.
[8] A. Ajorlou, and A. G. Aghdam, “Connectivity preservation in nonholonomic
multi-agent systems: a bounded distributed control strategy,” IEEE Transactions
on Automatic Control, vol. 58, no. 9, pp. 2366-2371, 2013.
[9] H. Ashrafiuon, S. Nersesov, and G. Clayton, “Trajectory tracking control of
planar underactuated vehicles,” IEEE Transactions on Automatic Control, no.
99, pp. 1-6, 2016.
[10] R. M. Murray, and S. S. Sastry, “Nonholonomic motion planning: steering
using sinusoids,” IEEE Transactions on Automatic Control, vol. 38, no. 5, pp.
700-716, 1993.
[11] Z. Peng, G. Wen, A. Rahmani, and Y. Yu, “Distributed consensus-based
formation control for multiple nonholonomic mobile robots with a specified
reference trajectory,” International Journal of Systems Science, vol. 46, no. 8,
pp. 1447-1457, 2015.
[12] Z. P. Jiang, and H. Nijmeijer, “A recursive technique for tracking control of
nonholonomic systems in chained form,” IEEE Transactions on Automatic
Control, vol. 44, no. 2, pp. 265-279, 1999.
[13] Z. Suna, S. S. Ge, W. Huo, and T. H. Lee, “Stabilization of nonholonomic
chained systems via nonregular feedback linearization,” Systems & Control
Letters, vol. 44, no. 4, pp. 279-289, 2001.
[14] Y. P. Tian, and K. C. Cao, “Time-varying linear controllers for exponential
tracking of non-holonomic systems in chained form,” International Journal
of Robust and Nonlinear Control, vol. 17, no. 7, pp. 631-647, 2007.
[15] W. Dong, and J. A. Farrell, “Cooperative control of multiple nonholonomic
mobile agents,” IEEE Transactions on Automatic Control, vol. 53, no. 6, pp.
1434-1448, 2008.
[16] W. Dong, “Distributed tracking control of networked chained systems,”
International Journal of Control, vol. 86, no. 12, pp. 2159-2174, 2013.
[17] A. Ferrara, L. Giacomini, and C. Vecchio, “Control of nonholonomic systems
with uncertainties via second-order sliding modes,” International Journal of
Robust and Nonlinear Control, vol. 18, no. 4-5, pp. 515-528, 2008.
[18] G. C.Walsh, and L. G. Bushnell, “Stabilization of multiple input chained form
control systems,” Proceeding of the 32nd IEEE Conference on Decision and
Control, pp. 959-964, 1993.
[19] Z. Li, J. Li, and Y. Kang, “Adaptive robust coordinated control of multiple
mobile manipulators interacting with rigid environments,” Automatica, vol. 46,
no. 12, pp. 2028-2034, 2010.
[20] M. Asif, M. J. Khan, and N. Cai, “Adaptive sliding mode dynamic controller
with integrator in the loop for nonholonomic wheeled mobile robot trajectory
tracking,” International Journal of Control, vol. 87, no. 5, pp. 964-975, 2014.
[21] J. Ye, “Tracking control of a non-holonomic wheeled mobile robot using
improved compound cosine function neural networks,” International Journal of
Control, vol. 88, no. 2, pp. 364-373, 2015.
[22] S. Delgado, and P. Kotyczka, “Energy shaping for position and speed control
of a wheeled inverted pendulum in reduced space,” Automatica, vol. 74, pp.
222-229, 2016.
[23] K. Shojaei, and A. M. Shahri, “Output feedback tracking control of uncertain
nonholonomic wheeled mobile robots: a dynamic surface control approach,”
IET Control Theory and Applications, vol. 6, no. 2, pp. 216-228, 2012.
[24] H. Gao, X. Song, L. Ding, K. Xia, N. Li, and Z. Deng, “Adaptive motion control
of wheeled mobile robot with unknown slippage,” International Journal of
Control, vol. 87, no. 8, pp. 1513-1522, 2014.
[25] J. Ghommam, H. Mehrjerdi, M. Saad, and F. Mnif, “Adaptive coordinated path
following control of non-holonomic mobile robots with quantised
communication,” IET Control Theory and Applications, vol. 5, no. 17, pp.
1990-2004, 2011.
[26] J. Fu, T. Chai, C. Y. Su, and Y. Jin, “Motion/force tracking control of
nonholonomic mechanical systems via combining cascaded design and
backstepping,” Automatica, vol. 49, no. 12, pp. 3682-3686, 2013.
[27] M. Oya, C. Y. Su, and R. Katoh, “Robust adaptive motion/force tracking
control of uncertain nonholonomic mechanical systems,” IEEE Transactions
on Robotics and Automation, vol. 19, no. 1, pp. 175-181, 2003.
[28] K. Shojaei, and A. M. Shahri, “Adaptive robust time-varying control of
uncertain non-holonomic robotic systems,” IET Control Theory and
Applications, vol. 6, no. 1, pp. 90-102, 2012.
[29] Z. Li, S. S. Ge, M. Adams, and W. S. Wijesoma, “Robust adaptive control of
uncertain force/motion constrained nonholonomic mobile manipulators,”
Automatica, vol. 44, no. 3, pp. 776-784, 2008.
[30] W. Dong, and J. A. Farrell, “Decentralized cooperative control of multiple
nonholonomic dynamic systems with uncertainty,” Automatica, vol. 45, no. 3,
pp. 706-710, 2009.
[31] H. Yuan, and Z. Qu, “Smooth time-varying pure feedback control for chained
nonholonomic systems with exponential convergent rate,” IET Control Theory
and Applications, vol. 4, no. 7, pp. 1235-1244, 2010.
[32] T. Kai, H. Kimura, and S. Hara, “Nonlinear control analysis on nonholonomic
dynamic systems with affine constraints,” Proceedings of the 44th IEEE
Conference on Decision and Control, and the European Control Conference
Seville, Spain, December 12-15, 2005.
[33] T. Kai, “Generalized canonical transformations and passivity-based control for
nonholonomic hamiltonian systems with affine constraints: control of a coin on
a rotating table,” Proceedings of the 46th IEEE Conference on Decision and
Control New Orleans, LA, USA, December 12-14, 2007.
[34] T. Kai, “Extended chained forms and their application to nonholonomic
kinematic systems with affine constraints: control of a coin on a rotating
table,” Proceedings of the 45th IEEE Conference on Decision and Control
Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006.
[35] T. Kai, “Derivation and analysis of nonholonomic hamiltonian systems with
affine constraints,” Proceedings of the European Control Conference 2007
Kos, Greece, July 2-5, 2007.
[36] W. Sun, Y. Q. Wu, and Z. Y. Sun, “Tracking control design for nonholonomic
mechanical systems with affine constraints,” International Journal of
Automation and Computing, vol. 11, no. 3, pp. 328-333, 2014.
[37] W. Sun, and Y. Wu, “Adaptive motion/force control of nonholonomic
mechanical systems with affine constraints,” Nonlinear Analysis: Modelling
and Control, vol. 19, no. 4, pp. 646-659, 2014.
[38] Z. Zhang, and Y. Wu, “Modeling and motion/force tracking for nonholonomic
dynamic systems with affine constraints,” 13th International Conference on
Control, Automation, Robotics & Vision Marina Bay Sands, Singapore, 10-12th
December, pp. 135-140, 2014.
[39] Z. Zhang, Y. Wu, and W. Sun, “Modeling and adaptive motion/force tracking
for vertical wheel on rotating table,” Journal of Systems Engineering and
Electronics, vol. 26, no. 5, pp. 1060-1069, 2015.
[40] T. Kai, “Modeling and passivity analysis of nonholonomic hamiltonian systems
with time-varying affine constraints,” Proceedings of the 18th IFAC World
Congress, 2011.
[41] C. -C. Cheng, S. -H. Chien, and F. -C. Shih, “Design of robust adaptive
variable structure tracking controllers with application to rigid robot
manipulators,” IET Control Theory and Applications, vol. 4, no. 9, pp. 1655-
1664, 2010.
[42] M.W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control,
One edition, New Jersey: John Wiley & Son, Inc., 2006.
[43] H. K. Khalil, Nonlinear Systems, Third edition, New Jersey: Prentice-Hall, Inc.,
2000.
[44] J. -J. E. Slotine,W. Li, Applied Nonlinear Control, New Jersey: Prentice-Hall,
Inc., 1991.
[45] S. J. Leon, Linear Algebra with Applications, 9th Edition, New Jersey:
Prentice-Hall, Inc., 2014.